3.399 \(\int \frac{(a+b x^3)^3 (c+d x+e x^2+f x^3+g x^4+h x^5)}{x^2} \, dx\)

Optimal. Leaf size=198 \[ \frac{1}{2} a^2 x^2 (a f+3 b c)+a^2 b d x^3+\frac{1}{4} a^2 x^4 (a h+3 b e)-\frac{a^3 c}{x}+a^3 d \log (x)+a^3 e x+\frac{1}{8} b^2 x^8 (3 a f+b c)+\frac{1}{2} a b^2 d x^6+\frac{1}{10} b^2 x^{10} (3 a h+b e)+\frac{3}{5} a b x^5 (a f+b c)+\frac{3}{7} a b x^7 (a h+b e)+\frac{g \left (a+b x^3\right )^4}{12 b}+\frac{1}{9} b^3 d x^9+\frac{1}{11} b^3 f x^{11}+\frac{1}{13} b^3 h x^{13} \]

[Out]

-((a^3*c)/x) + a^3*e*x + (a^2*(3*b*c + a*f)*x^2)/2 + a^2*b*d*x^3 + (a^2*(3*b*e + a*h)*x^4)/4 + (3*a*b*(b*c + a
*f)*x^5)/5 + (a*b^2*d*x^6)/2 + (3*a*b*(b*e + a*h)*x^7)/7 + (b^2*(b*c + 3*a*f)*x^8)/8 + (b^3*d*x^9)/9 + (b^2*(b
*e + 3*a*h)*x^10)/10 + (b^3*f*x^11)/11 + (b^3*h*x^13)/13 + (g*(a + b*x^3)^4)/(12*b) + a^3*d*Log[x]

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Rubi [A]  time = 0.182106, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {1583, 1820} \[ \frac{1}{2} a^2 x^2 (a f+3 b c)+a^2 b d x^3+\frac{1}{4} a^2 x^4 (a h+3 b e)-\frac{a^3 c}{x}+a^3 d \log (x)+a^3 e x+\frac{1}{8} b^2 x^8 (3 a f+b c)+\frac{1}{2} a b^2 d x^6+\frac{1}{10} b^2 x^{10} (3 a h+b e)+\frac{3}{5} a b x^5 (a f+b c)+\frac{3}{7} a b x^7 (a h+b e)+\frac{g \left (a+b x^3\right )^4}{12 b}+\frac{1}{9} b^3 d x^9+\frac{1}{11} b^3 f x^{11}+\frac{1}{13} b^3 h x^{13} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^3)^3*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/x^2,x]

[Out]

-((a^3*c)/x) + a^3*e*x + (a^2*(3*b*c + a*f)*x^2)/2 + a^2*b*d*x^3 + (a^2*(3*b*e + a*h)*x^4)/4 + (3*a*b*(b*c + a
*f)*x^5)/5 + (a*b^2*d*x^6)/2 + (3*a*b*(b*e + a*h)*x^7)/7 + (b^2*(b*c + 3*a*f)*x^8)/8 + (b^3*d*x^9)/9 + (b^2*(b
*e + 3*a*h)*x^10)/10 + (b^3*f*x^11)/11 + (b^3*h*x^13)/13 + (g*(a + b*x^3)^4)/(12*b) + a^3*d*Log[x]

Rule 1583

Int[(Px_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(Coeff[Px, x, n - m - 1]*(a + b*x^n)^(p
+ 1))/(b*n*(p + 1)), x] + Int[(Px - Coeff[Px, x, n - m - 1]*x^(n - m - 1))*x^m*(a + b*x^n)^p, x] /; FreeQ[{a,
b, m, n}, x] && PolyQ[Px, x] && IGtQ[p, 1] && IGtQ[n - m, 0] && NeQ[Coeff[Px, x, n - m - 1], 0]

Rule 1820

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a +
b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rubi steps

\begin{align*} \int \frac{\left (a+b x^3\right )^3 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{x^2} \, dx &=\frac{g \left (a+b x^3\right )^4}{12 b}+\int \frac{\left (a+b x^3\right )^3 \left (c+d x+e x^2+f x^3+h x^5\right )}{x^2} \, dx\\ &=\frac{g \left (a+b x^3\right )^4}{12 b}+\int \left (a^3 e+\frac{a^3 c}{x^2}+\frac{a^3 d}{x}+a^2 (3 b c+a f) x+3 a^2 b d x^2+a^2 (3 b e+a h) x^3+3 a b (b c+a f) x^4+3 a b^2 d x^5+3 a b (b e+a h) x^6+b^2 (b c+3 a f) x^7+b^3 d x^8+b^2 (b e+3 a h) x^9+b^3 f x^{10}+b^3 h x^{12}\right ) \, dx\\ &=-\frac{a^3 c}{x}+a^3 e x+\frac{1}{2} a^2 (3 b c+a f) x^2+a^2 b d x^3+\frac{1}{4} a^2 (3 b e+a h) x^4+\frac{3}{5} a b (b c+a f) x^5+\frac{1}{2} a b^2 d x^6+\frac{3}{7} a b (b e+a h) x^7+\frac{1}{8} b^2 (b c+3 a f) x^8+\frac{1}{9} b^3 d x^9+\frac{1}{10} b^2 (b e+3 a h) x^{10}+\frac{1}{11} b^3 f x^{11}+\frac{1}{13} b^3 h x^{13}+\frac{g \left (a+b x^3\right )^4}{12 b}+a^3 d \log (x)\\ \end{align*}

Mathematica [A]  time = 0.136937, size = 172, normalized size = 0.87 \[ \frac{1}{140} a^2 b x^2 \left (210 c+x \left (140 d+x \left (105 e+84 f x+70 g x^2+60 h x^3\right )\right )\right )+a^3 \left (-\frac{c}{x}+e x+\frac{1}{12} x^2 \left (6 f+4 g x+3 h x^2\right )\right )+a^3 d \log (x)+\frac{1}{840} a b^2 x^5 \left (504 c+x \left (420 d+x \left (360 e+315 f x+280 g x^2+252 h x^3\right )\right )\right )+\frac{b^3 x^8 \left (6435 c+5720 d x+6 x^2 \left (858 e+780 f x+715 g x^2+660 h x^3\right )\right )}{51480} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^3)^3*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/x^2,x]

[Out]

a^3*(-(c/x) + e*x + (x^2*(6*f + 4*g*x + 3*h*x^2))/12) + (b^3*x^8*(6435*c + 5720*d*x + 6*x^2*(858*e + 780*f*x +
715*g*x^2 + 660*h*x^3)))/51480 + (a^2*b*x^2*(210*c + x*(140*d + x*(105*e + 84*f*x + 70*g*x^2 + 60*h*x^3))))/1
40 + (a*b^2*x^5*(504*c + x*(420*d + x*(360*e + 315*f*x + 280*g*x^2 + 252*h*x^3))))/840 + a^3*d*Log[x]

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Maple [A]  time = 0.007, size = 224, normalized size = 1.1 \begin{align*}{\frac{{b}^{3}h{x}^{13}}{13}}+{\frac{{b}^{3}g{x}^{12}}{12}}+{\frac{{b}^{3}f{x}^{11}}{11}}+{\frac{3\,{x}^{10}a{b}^{2}h}{10}}+{\frac{{b}^{3}e{x}^{10}}{10}}+{\frac{{x}^{9}a{b}^{2}g}{3}}+{\frac{{b}^{3}d{x}^{9}}{9}}+{\frac{3\,{x}^{8}a{b}^{2}f}{8}}+{\frac{{b}^{3}c{x}^{8}}{8}}+{\frac{3\,{x}^{7}{a}^{2}bh}{7}}+{\frac{3\,a{b}^{2}e{x}^{7}}{7}}+{\frac{{x}^{6}{a}^{2}bg}{2}}+{\frac{a{b}^{2}d{x}^{6}}{2}}+{\frac{3\,{x}^{5}{a}^{2}bf}{5}}+{\frac{3\,a{b}^{2}c{x}^{5}}{5}}+{\frac{{x}^{4}{a}^{3}h}{4}}+{\frac{3\,{a}^{2}be{x}^{4}}{4}}+{\frac{{x}^{3}{a}^{3}g}{3}}+{a}^{2}bd{x}^{3}+{\frac{{a}^{3}f{x}^{2}}{2}}+{\frac{3\,{a}^{2}bc{x}^{2}}{2}}+{a}^{3}ex+{a}^{3}d\ln \left ( x \right ) -{\frac{{a}^{3}c}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^3*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/x^2,x)

[Out]

1/13*b^3*h*x^13+1/12*b^3*g*x^12+1/11*b^3*f*x^11+3/10*x^10*a*b^2*h+1/10*b^3*e*x^10+1/3*x^9*a*b^2*g+1/9*b^3*d*x^
9+3/8*x^8*a*b^2*f+1/8*b^3*c*x^8+3/7*x^7*a^2*b*h+3/7*a*b^2*e*x^7+1/2*x^6*a^2*b*g+1/2*a*b^2*d*x^6+3/5*x^5*a^2*b*
f+3/5*a*b^2*c*x^5+1/4*x^4*a^3*h+3/4*a^2*b*e*x^4+1/3*x^3*a^3*g+a^2*b*d*x^3+1/2*a^3*f*x^2+3/2*a^2*b*c*x^2+a^3*e*
x+a^3*d*ln(x)-a^3*c/x

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Maxima [A]  time = 0.945867, size = 286, normalized size = 1.44 \begin{align*} \frac{1}{13} \, b^{3} h x^{13} + \frac{1}{12} \, b^{3} g x^{12} + \frac{1}{11} \, b^{3} f x^{11} + \frac{1}{10} \,{\left (b^{3} e + 3 \, a b^{2} h\right )} x^{10} + \frac{1}{9} \,{\left (b^{3} d + 3 \, a b^{2} g\right )} x^{9} + \frac{1}{8} \,{\left (b^{3} c + 3 \, a b^{2} f\right )} x^{8} + \frac{3}{7} \,{\left (a b^{2} e + a^{2} b h\right )} x^{7} + \frac{1}{2} \,{\left (a b^{2} d + a^{2} b g\right )} x^{6} + \frac{3}{5} \,{\left (a b^{2} c + a^{2} b f\right )} x^{5} + a^{3} e x + \frac{1}{4} \,{\left (3 \, a^{2} b e + a^{3} h\right )} x^{4} + a^{3} d \log \left (x\right ) + \frac{1}{3} \,{\left (3 \, a^{2} b d + a^{3} g\right )} x^{3} - \frac{a^{3} c}{x} + \frac{1}{2} \,{\left (3 \, a^{2} b c + a^{3} f\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^3*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/x^2,x, algorithm="maxima")

[Out]

1/13*b^3*h*x^13 + 1/12*b^3*g*x^12 + 1/11*b^3*f*x^11 + 1/10*(b^3*e + 3*a*b^2*h)*x^10 + 1/9*(b^3*d + 3*a*b^2*g)*
x^9 + 1/8*(b^3*c + 3*a*b^2*f)*x^8 + 3/7*(a*b^2*e + a^2*b*h)*x^7 + 1/2*(a*b^2*d + a^2*b*g)*x^6 + 3/5*(a*b^2*c +
a^2*b*f)*x^5 + a^3*e*x + 1/4*(3*a^2*b*e + a^3*h)*x^4 + a^3*d*log(x) + 1/3*(3*a^2*b*d + a^3*g)*x^3 - a^3*c/x +
1/2*(3*a^2*b*c + a^3*f)*x^2

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Fricas [A]  time = 1.02106, size = 570, normalized size = 2.88 \begin{align*} \frac{27720 \, b^{3} h x^{14} + 30030 \, b^{3} g x^{13} + 32760 \, b^{3} f x^{12} + 36036 \,{\left (b^{3} e + 3 \, a b^{2} h\right )} x^{11} + 40040 \,{\left (b^{3} d + 3 \, a b^{2} g\right )} x^{10} + 45045 \,{\left (b^{3} c + 3 \, a b^{2} f\right )} x^{9} + 154440 \,{\left (a b^{2} e + a^{2} b h\right )} x^{8} + 180180 \,{\left (a b^{2} d + a^{2} b g\right )} x^{7} + 216216 \,{\left (a b^{2} c + a^{2} b f\right )} x^{6} + 360360 \, a^{3} e x^{2} + 90090 \,{\left (3 \, a^{2} b e + a^{3} h\right )} x^{5} + 360360 \, a^{3} d x \log \left (x\right ) + 120120 \,{\left (3 \, a^{2} b d + a^{3} g\right )} x^{4} - 360360 \, a^{3} c + 180180 \,{\left (3 \, a^{2} b c + a^{3} f\right )} x^{3}}{360360 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^3*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/x^2,x, algorithm="fricas")

[Out]

1/360360*(27720*b^3*h*x^14 + 30030*b^3*g*x^13 + 32760*b^3*f*x^12 + 36036*(b^3*e + 3*a*b^2*h)*x^11 + 40040*(b^3
*d + 3*a*b^2*g)*x^10 + 45045*(b^3*c + 3*a*b^2*f)*x^9 + 154440*(a*b^2*e + a^2*b*h)*x^8 + 180180*(a*b^2*d + a^2*
b*g)*x^7 + 216216*(a*b^2*c + a^2*b*f)*x^6 + 360360*a^3*e*x^2 + 90090*(3*a^2*b*e + a^3*h)*x^5 + 360360*a^3*d*x*
log(x) + 120120*(3*a^2*b*d + a^3*g)*x^4 - 360360*a^3*c + 180180*(3*a^2*b*c + a^3*f)*x^3)/x

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Sympy [A]  time = 0.633058, size = 236, normalized size = 1.19 \begin{align*} - \frac{a^{3} c}{x} + a^{3} d \log{\left (x \right )} + a^{3} e x + \frac{b^{3} f x^{11}}{11} + \frac{b^{3} g x^{12}}{12} + \frac{b^{3} h x^{13}}{13} + x^{10} \left (\frac{3 a b^{2} h}{10} + \frac{b^{3} e}{10}\right ) + x^{9} \left (\frac{a b^{2} g}{3} + \frac{b^{3} d}{9}\right ) + x^{8} \left (\frac{3 a b^{2} f}{8} + \frac{b^{3} c}{8}\right ) + x^{7} \left (\frac{3 a^{2} b h}{7} + \frac{3 a b^{2} e}{7}\right ) + x^{6} \left (\frac{a^{2} b g}{2} + \frac{a b^{2} d}{2}\right ) + x^{5} \left (\frac{3 a^{2} b f}{5} + \frac{3 a b^{2} c}{5}\right ) + x^{4} \left (\frac{a^{3} h}{4} + \frac{3 a^{2} b e}{4}\right ) + x^{3} \left (\frac{a^{3} g}{3} + a^{2} b d\right ) + x^{2} \left (\frac{a^{3} f}{2} + \frac{3 a^{2} b c}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**3*(h*x**5+g*x**4+f*x**3+e*x**2+d*x+c)/x**2,x)

[Out]

-a**3*c/x + a**3*d*log(x) + a**3*e*x + b**3*f*x**11/11 + b**3*g*x**12/12 + b**3*h*x**13/13 + x**10*(3*a*b**2*h
/10 + b**3*e/10) + x**9*(a*b**2*g/3 + b**3*d/9) + x**8*(3*a*b**2*f/8 + b**3*c/8) + x**7*(3*a**2*b*h/7 + 3*a*b*
*2*e/7) + x**6*(a**2*b*g/2 + a*b**2*d/2) + x**5*(3*a**2*b*f/5 + 3*a*b**2*c/5) + x**4*(a**3*h/4 + 3*a**2*b*e/4)
+ x**3*(a**3*g/3 + a**2*b*d) + x**2*(a**3*f/2 + 3*a**2*b*c/2)

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Giac [A]  time = 1.06084, size = 308, normalized size = 1.56 \begin{align*} \frac{1}{13} \, b^{3} h x^{13} + \frac{1}{12} \, b^{3} g x^{12} + \frac{1}{11} \, b^{3} f x^{11} + \frac{3}{10} \, a b^{2} h x^{10} + \frac{1}{10} \, b^{3} x^{10} e + \frac{1}{9} \, b^{3} d x^{9} + \frac{1}{3} \, a b^{2} g x^{9} + \frac{1}{8} \, b^{3} c x^{8} + \frac{3}{8} \, a b^{2} f x^{8} + \frac{3}{7} \, a^{2} b h x^{7} + \frac{3}{7} \, a b^{2} x^{7} e + \frac{1}{2} \, a b^{2} d x^{6} + \frac{1}{2} \, a^{2} b g x^{6} + \frac{3}{5} \, a b^{2} c x^{5} + \frac{3}{5} \, a^{2} b f x^{5} + \frac{1}{4} \, a^{3} h x^{4} + \frac{3}{4} \, a^{2} b x^{4} e + a^{2} b d x^{3} + \frac{1}{3} \, a^{3} g x^{3} + \frac{3}{2} \, a^{2} b c x^{2} + \frac{1}{2} \, a^{3} f x^{2} + a^{3} x e + a^{3} d \log \left ({\left | x \right |}\right ) - \frac{a^{3} c}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^3*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/x^2,x, algorithm="giac")

[Out]

1/13*b^3*h*x^13 + 1/12*b^3*g*x^12 + 1/11*b^3*f*x^11 + 3/10*a*b^2*h*x^10 + 1/10*b^3*x^10*e + 1/9*b^3*d*x^9 + 1/
3*a*b^2*g*x^9 + 1/8*b^3*c*x^8 + 3/8*a*b^2*f*x^8 + 3/7*a^2*b*h*x^7 + 3/7*a*b^2*x^7*e + 1/2*a*b^2*d*x^6 + 1/2*a^
2*b*g*x^6 + 3/5*a*b^2*c*x^5 + 3/5*a^2*b*f*x^5 + 1/4*a^3*h*x^4 + 3/4*a^2*b*x^4*e + a^2*b*d*x^3 + 1/3*a^3*g*x^3
+ 3/2*a^2*b*c*x^2 + 1/2*a^3*f*x^2 + a^3*x*e + a^3*d*log(abs(x)) - a^3*c/x